Order of variables in a Jacobian?

[peripatein]

Hi,

Homework Statement

I was hoping someone could please explain the order of variables in a Jacobian. I mean, once the dependent and independent variables have been identified, how should the Jacobian be formulated. For instance, supposing I have two implicit functions F(x,y,u,v) and G(x,y,u,v) where x and y are independent. I wish to find ∂u/∂y (x is fixed) and ∂v/∂y (x is fixed). How should the Jacobian be formulated, namely -J[(F,G)/(y,u)]/J[(F,G)/(v,u)] or -J[(F,G)/(u,y)]/J[(F,G)/(u,v)]? I happen to know the former formulation is the correct one, but why?! How could I have known and initially written it in that form?

Homework Equations

The Attempt at a Solution

 

[Dustinsfl]

Consider two functions $f$ and $g$ of two variables y and x.
I always form my Jacobian like
$$
\mathcal{J} = \begin{pmatrix}
f_x & f_y\\
g_x & g_y
\end{pmatrix}
$$
However, I believe in some books it has my rows as columns.

 

[Dick]

What's usually important is the absolute value of the determinant of the jacobian matrix. In which case none of these variations matter.

 

[Ray Vickson]

Hi,

Homework Statement

I was hoping someone could please explain the order of variables in a Jacobian. I mean, once the dependent and independent variables have been identified, how should the Jacobian be formulated. For instance, supposing I have two implicit functions F(x,y,u,v) and G(x,y,u,v) where x and y are independent. I wish to find ∂u/∂y (x is fixed) and ∂v/∂y (x is fixed). How should the Jacobian be formulated, namely -J[(F,G)/(y,u)]/J[(F,G)/(v,u)] or -J[(F,G)/(u,y)]/J[(F,G)/(u,v)]? I happen to know the former formulation is the correct one, but why?! How could I have known and initially written it in that form?

Homework Equations

The Attempt at a Solution

You can always get it from first principles (and doing this once in your life is a useful exercise). If we fix y and let x change by Δx = h, then we have:
$$F(x+h,y,u + \Delta u, v + \Delta v) = 0 = F(x,y,u,v) + F_x h + F_u \Delta u + F_v \Delta v \\ G(x+h,y,u + \Delta u, v + \Delta v) = 0 = G(x,y,u,v) + G_x h + G_u \Delta u + G_v \Delta v$$
where all the partials are evaluated at the original point (x,y,u,v). Thus,
$$\pmatrix{ \Delta u \\ \Delta v} = - \pmatrix{F_u & F_v\\G_u&G_v}^{-1}\pmatrix{F_x \\ G_x} h,$$
so
$$\pmatrix{\partial u/ \partial x \\ \partial v / \partial x} = - \pmatrix{F_u & F_v\\G_u&G_v}^{-1}\pmatrix{F_x \\ G_x}.$$
For a 2x2 matrix we get the inverse by swapping the diagonal elements, changing the sign of the off-diagonal elements and dividing by the determinant:
$$\pmatrix{a & b \\ c & d }^{-1} = \frac{1}{ad-bc} \pmatrix{d & -b \\ -c & a},$$ so you can get explicit formulas for u_x and v_x.